Analysis of results Directory UMM :Journals:Journal of Operations Management:Vol18.Issue4.Jun2000:

eries, and the number of patients treated and re- jected.

4. Analysis of results

Table 6 presents the results for the DICU. The numbers of patients treated and rejected are compa- rable to the current ICU, except for OT-electives. More DICU beds mean fewer cancelled surgeries and lower bed utilization for OT-electives. Other than patients treated, all performance measures dete- riorate for the other groups. This is not unexpected since increased DICU capacity means decreased ca- pacity for the other groups. The issue, then, is not whether a DICU impacts adversely on other patients. Rather, the issue is whether any favourable effects on ES patients outweigh those adverse impacts. In no case did a DICU outperform the current system in all aspects. The three-bed unit performed comparably in terms of bed utilization, but cancelled surgeries increased fivefold. With a six-bed DICU, cancelled surgeries more than halved, at a cost of a significant deterioration in queue performance for the main group. Specifically, the average number of patients in the queue increased from 0.04 to 2.40. In sum, a DICU only resolves the cancelled-surgeries problem by imposing substantially poorer service on the other patients. FBA is evaluated with n of the 13 ICU beds reserved for ES patients, with n increased from 1 to Table 6 Evaluation of the performance of the dependency unit Weighted by the number of patients in each source. Ž . The average times in queue for the three groups are not statistically different a s 0.10 . The admission rule is FCFS and all three groups are waiting in the same queue. Performance measures Current DICU 3 DICU 4 DICU 5 DICU 6 Ž . Bed utilization r Main 0.5958 0.5816 0.6460 0.7263 0.8270 DICU 0.5049 0.4405 0.3814 0.3305 Ž . Average no. of patients in queue L 0.04 0.11 0.27 0.76 2.40 q 1 Ž . Average time in queue W 0.37 h 1.11 2.84 7.94 24.96 q Ward 0.39 1.13 2.88 7.97 25.11 A E 0.36 1.03 2.76 7.98 24.97 OT-emergency 0.34 1.16 2.84 7.79 24.48 OT-elective – – – – – Ž . Average time in system W Ward 63.59 64.33 66.08 71.17 88.28 A E 49.28 49.95 51.68 56.90 73.88 OT-emergency 75.65 76.47 78.15 83.10 99.75 OT-elective 47.94 47.65 47.70 47.75 47.75 No. of cancelled surgeries 18.10 92.75 48.05 21.85 8.10 No. of patients treatedryr Ward 394.1 394.1 394.1 394.1 394.1 A E 264.8 264.8 264.8 264.8 264.8 OT-emergency 173.3 173.3 173.3 173.3 173.3 OT-elective 354.3 279.7 324.4 350.6 364.3 No. of patients rejectedryr Ward 386.5 386.5 386.5 386.5 386.5 A E 117.3 117.3 117.3 117.3 117.3 OT-emergency 6.1 6.1 6.1 6.1 6.1 OT-elective 10.2 10.2 10.2 10.2 10.2 5 incrementally. As anticipated, as n increases, the measures for the favoured ES patients improve while those for the others worsen. Specifically, cancelled surgeries again decline, but for the other groups the waiting times and the number of waiting patients increase. Once again the issue is whether the benefits to the favoured group offset the costs to the others. The changes for these measures are nonlinear functions of the number of beds reserved. As more beds are reserved, the average waiting time in- creases, and cancelled surgeries decrease, at higher rates. Among the five cases tested, for expositional purposes the three-bed case was chosen for our main experiment as it gives the greatest reduction in can- celled surgeries with relatively less effect on the average queue size. This choice reflects our assess- ment of the alternatives and the performance criteria. Table 7 shows the results of the pilot runs for the exclusive-reservation method. Table 8 compares the four FBA bed-reservation schemes with the current ICU. Three beds are re- served in all four cases. As with a DICU, all four schemes produced numbers of patients treated and rejected that are comparable to the current numbers. The four schemes, however, resulted in different levels of performance for the criteria of primary interest: average queue size and queuing times for patients coming from the Ward, A E, and OT- emergency, and the number of cancelled surgeries. It is immediately apparent ex post that our ex ante conjectures are realized. On the one hand, a reserva- tion system benefits the surgeons and ES patients, insofar as it reduces the number of cancelled surg- eries. On the other hand, the system works to the Table 7 Ž . Determining the optimal number of beds for FBA scheme exclusive reservation Ž . The average times in queue for the three groups are not statistically different a s 0.10 . The admission rule is FCFS and all three groups are waiting in the same queue. Weighted by the number of patients in each source. Ž . Ž . Ž . Ž . Ž . Performance measures FBA 1 FBA 2 FBA 3 FBA 4 FBA 5 Ž . Bed utilization r 0.5962 0.5968 0.5977 0.5986 0.6000 Ž . Average no. of patients in queue L 0.05 0.08 0.15 0.31 0.79 q Ž . Average time in queue W 0.55 0.84 1.57 3.27 8.24 q Ward 0.57 0.89 1.62 3.31 8.26 A E 0.53 0.78 1.46 3.19 8.25 OT-emergency 0.53 0.81 1.59 3.28 8.15 OT-elective – – – – – Ž . Average time in system W Ward 63.77 64.09 64.82 66.51 71.46 A E 49.45 49.71 50.39 52.11 57.17 OT-emergency 75.84 76.12 76.90 78.59 83.45 OT-elective 47.79 47.83 47.94 47.97 47.90 No. of cancelled surgeries 16.75 15.35 13.65 11.30 8.05 No. of patients treatedryr Ward 394.1 394.1 394.1 394.1 394.1 A E 264.8 264.8 264.8 264.8 264.8 OT-emergency 173.3 173.3 173.3 173.3 173.3 OT-elective 355.7 357.1 358.8 361.1 364.4 No. of patients rejectedryr Ward 386.5 386.5 386.5 386.5 386.5 A E 117.3 117.3 117.3 117.3 117.3 OT-emergency 6.1 6.1 6.1 6.1 6.1 OT-elective 10.2 10.2 10.2 10.2 10.2 Table 8 Comparison of four bed reservation schemes of FBA Ž . The average times in queue for the three groups are not statistically different a s 0.10 except for special arrangements such as Open to AE. Weighted by the number of patients in each source. Performance measures Current Exclusive Open to Open to Share with Ž . Ž . AE FrSat all FrirSat OTM Ž . Bed utilization r 0.5958 0.5977 0.5975 0.5972 0.5970 Ž . Average no. of patients in queue L 0.04 0.15 0.10 0.07 0.06 q Ž . Average time in queue W 0.37 h 1.57 1.04 0.76 0.65 q Ward 0.39 1.62 1.17 0.81 0.75 A E 0.36 1.46 0.79 0.71 0.71 OT-emergency 0.34 1.59 1.09 0.68 0.30 OT-elective – – – – – Ž . Average time in system W Ward 63.59 64.82 64.37 64.01 63.95 A E 49.28 50.39 49.72 49.64 49.63 OT-emergency 75.65 76.90 76.40 75.99 75.61 OT-elective 47.94 47.94 47.95 47.96 47.96 No. of cancelled surgeries 18.10 13.65 14.45 15.30 16.30 No. of patients treatedryr Ward 394.1 394.1 394.1 394.1 394.1 A E 264.8 264.8 264.8 264.8 264.8 OT-emergency 173.3 173.3 173.3 173.3 173.3 OT-elective 354.3 358.8 358.0 357.1 356.1 No. patients rejectedryr Ward 386.5 386.5 386.5 386.5 386.5 A E 117.3 117.3 117.3 117.3 117.3 OT-emergency 6.1 6.1 6.1 6.1 6.1 OT-elective 10.2 10.2 10.2 10.2 10.2 detriment of all other patients, insofar as it lengthens their average waiting times. Put otherwise, there is no win–win situation here. Instead, tradeoffs have to be made and evaluated. The ultimate responsibility for making that evaluation and persuading the physi- cians and surgeons to accept the system hershe chooses falls to the administrator. Notwithstanding the different tradeoffs, bed uti- lization under all four FBA variants is only marginally higher than with the current system. The most profound differences are in the average queuing times and cancelled surgeries. The exclusivity of Strategy 2 gives the greatest reduction in cancelled surgeries, a 24.6 reduction from 18.10 to 13.65. The cost is an average of 1.2 additional hours in queuing time that is borne almost uniformly by patients from each of the other sources. That addi- tional time can be halved by Strategy 3, opening the reserved beds to A E patients on Friday and Satur- day, but at a price of another cancelled surgery. This is a 6 increase in cancelled surgeries over the number with Strategy 2. We would have preferred our results to contain at least one patently superior option that markedly reduces cancelled elective surg- eries and has a negligible impact on waiting times. Such an option is unavailable. Still, our results have two important managerial implications. First, our results support the contention that an exclusive reservation FBA system will markedly re- duce cancelled surgeries. Strategy 2 has merit when 1.2 additional hours in a queue will not be seriously detrimental to a patient’s physical andror mental condition. This additional waiting time is the only cost of FBA. By the same token, if the increased waiting time is expected to be detrimental to the patient, our results provide the administrator the ammunition with which to fend off the surgeons’ requests to implement some form of FBA. Second, insofar as some reservation system is to be put in place, it is apparent that FBA has several advantages over DICU. The first is that FBA offers greater flexibility in resource utilization; the two units can share such resources as doctors and nurses without additional monetary cost. A second FBA advantage is its greater flexibility in bed allocation. The number of beds reserved can be readily changed if the patient arrival numbers change, as might occur between different seasons. Third, FBA will be more effective in offsetting the demand fluctuations be- tween groups. This is achieved by merging several groups into the same unit as opposed to isolating them in a DICU. Finally, our results indicate that FBA performs better on similar dimensions with a smaller number of reserved beds, and without the higher operating costs required for a DICU to per- form comparably. 4.1. The efficient frontier As shown in Table 9, the DICU option and all four FBA schemes result in statistically significant Ž . p - 0.004 increases in waiting times for the non- favoured groups. Statistical significance, however, need not imply that the increases are important in terms of how their impact on the patients’ health and welfare. As Table 9 also reveals two FBA schemes Ž . produce statistically significant a s 0.10 decreases in cancelled surgeries. These decreases may be im- portant, too. In evaluating alternatives, the adminis- trator must do a balancing act between statistically significant tradeoffs involving the two conflicting Table 9 Pair-wise comparison of the means for the difference between the current system and the DICU and FBA schemes p-Values for one-tailed tests are shown in the parenthesis. Systems Number of Waiting time cancelled surgeries Current 18.10 0.37 Ž . Ž . DICU 92.75 0.0000 1.11 0.0001 Ž . Ž . FBA-exclusive 13.65 0.0219 1.57 0.0000 Ž . Ž . FBA-open to AE 14.45 0.0547 1.04 0.0000 Ž . Ž . FBA-open to all 15.30 0.1181 0.76 0.0005 Ž . Ž . FBA-share with 16.30 0.2293 0.65 0.0037 OT-emergency objectives of greatest moment to himrher in a multi- ple-objective problem. As an interesting aside, although the performance numbers differed, essentially the same inferences were reached in our initial experiments when we generated the OT-elective service times using an exponential density whose parameter was taken as Ž . the standard deviation 65.78 for that category from the empirical data of Table 4. We chose to use the standard deviation rather than the mean, in order to be conservative in light of our rejection of the expo- nential-density hypothesis. The fact that the same policy inferences are drawn from such widely differ- Ž ing assumptions is not surprising. As Baker 1974, p. . 230 wrote a quarter of a century ago, ‘‘the implica- Ž . tion that a policy does not suffer drastically if input information is unreliable suggests that the simplifica- tions often made in experimental models may not be crucial to the adaptation of simulation results to actual shop control’’. In a related vein, rather than reducing the number of beds, we considered modelling the service times of the outliers, but rejected this option as there are only 32 patients in our set, with 15 coming from the Ward. Their fewness in numbers and the fact their service times are spread all over the map, is what makes them outliers. The only density that would seem appropriate for modelling these outliers is a uniform density, we would require a unique density for each of the four patient sources, and no case has a sufficient number of observations to permit a good- ness-of-fit test. Rather than invoking one of these suspect densities with probability 0.0589, we have chosen to reduce the number of beds by 7.14. The numbers that we get might therefore be somewhat different from those in the alternative procedure, but they are conservative numbers that will yield the same basic conclusions and policy inferences as in the alternative approach. Returning to our main theme, a classic approach for dealing with the multiple-objective problem is to determine the efficient frontier or trade-off curve that summarizes the Pareto-optimal tradeoffs between Ž these objectives Winston and Albright, 1997, pp. . 353–354 . This approach has previously been used Ž . in a hospital setting Banker et al., 1986 , as well as in a variety of other settings. These include financial ŽGollinger and Morgan, 1993; Herrick, 1997; Morey Fig. 5. The efficient frontier. . Ž and Morey, 1999 , managerial Thanassoulis et al., . Ž 1996 , industrial Park and Simar, 1994; Tamiz et . Ž al., 1999 , economics Sueyoshi et al., 1998; Hibiki Ž . Fig. 6. The effects of adding one more bed 14 beds . . Ž and Sueyoshi, 1999 , marketing Horsky and Nelson, . Ž . 1996 , and production De et al., 1992 settings. Fig. Table 10 The effect of increasing the number of beds by 1 to 14 Weighted by the number of patients in each source. Ž . The average times in queue for the three groups are not statistically different a s 0.10 except for special arrangements such as Open to AE. Performance measures Current Exclusive Open to Open to Share with Ž . Ž . AE FrirSat FrirSat OTM Ž . Bed utilization r 0.5565 0.5574 0.5573 0.5571 0.5570 Ž . Average no. of patients in queue L 0.01 0.06 0.04 0.03 0.02 q Ž . Average time in queue W 0.14 h 0.64 0.42 0.29 0.24 q Ward 0.15 0.68 0.47 0.30 0.29 A E 0.14 0.57 0.31 0.28 0.26 OT-emergency 0.14 0.66 0.48 0.26 0.12 OT-elective – – – – – Ž . Average time in system W Ward 63.35 63.88 63.67 63.51 63.49 A E 49.06 49.49 49.23 49.20 49.19 OT-emergency 75.45 75.97 75.79 75.57 75.42 OT-elective 47.80 47.82 47.81 47.89 47.86 No. of cancelled surgeries 9.60 7.70 7.95 8.85 8.90 No. of patients treatedryr Ward 394.1 394.1 394.1 394.1 394.1 A E 264.8 264.8 264.8 264.8 264.8 OT-emergency 173.3 173.3 173.3 173.3 173.3 OT-elective 362.8 364.7 364.5 363.6 363.5 No. of patients rejectedryr Ward 386.5 386.5 386.5 386.5 386.5 A E 117.3 117.3 117.3 117.3 117.3 OT-emergency 6.1 6.1 6.1 6.1 6.1 OT-elective 10.2 10.2 10.2 10.2 10.2 Fig. 7. Number of admitted patients by month over the 2-yr period. 5 presents the efficient frontier that can be achieved through the current system and the four FBA bed-re- servation schemes. The dots indicate the five points determined from the data of Table 8. The curve is sketched from four of those points; the fifth point, which is associated with opening the reserved beds to OT-emergency patients throughout the week, sits above the curve. The efficient frontier has a negative slope and is convex. The slope is negative, because any reduction in the number of cancelled surgeries comes at the expense of additional waiting time. The frontier is necessarily convex, because we can always achieve a point on the straight line connecting any two points by a linear combination of those points. That is, we can employ a mixed strategy that implements one of the latter two pure strategies x percent of the time Ž . and the other 1 y x percent of the time. The strictly concave curves are hypothetical indif- ference curves drawn solely for illustrative purposes. Table 11 Evaluation of the impacts of increased arrival rates with 40 increase Ž . The average times in queue for the three groups are not statistically different a s 0.10 except for special arrangements such as Open to AE. Weighted by the number of patients in each source. Performance measures Current Exclusive Open to Open to all Share with Ž . Ž . AE FrirSat FrirSat OTM Ž . Bed utilization r 0.7903 0.8000 0.7990 0.7997 0.7930 Ž . Average no. of patients in queue L 0.46 2.35 1.34 1.11 0.80 q Ž . Average time in queue W 3.43 h 17.49 9.98 8.28 5.95 q Ward 3.43 17.51 10.68 8.31 6.78 A E 3.53 17.71 8.72 8.33 6.85 OT-emergency 3.29 17.12 10.33 8.14 2.61 OT-elective – – – – – Ž . Average time in system W Ward 66.90 80.98 74.15 71.78 70.25 A E 52.57 66.73 57.75 57.36 55.88 OT-emergency 78.48 92.29 85.51 83.33 77.80 OT-elective 49.12 48.32 48.74 48.94 48.99 No. of cancelled surgeries 155.90 114.00 123.00 130.05 143.85 No. of patients treatedryr Ward 555.2 555.2 555.2 555.2 555.2 A E 375.5 375.5 375.5 375.5 375.5 OT-emergency 242.2 242.2 242.2 242.2 242.2 OT-elective 373.4 415.3 406.3 399.2 385.4 No. patients rejectedryr Ward 535.6 535.6 535.6 535.6 535.6 A E 162.1 162.1 162.1 162.1 162.1 OT-emergency 8.3 8.3 8.3 8.3 8.3 OT-elective 14.8 14.8 14.8 14.8 14.8 They are strictly concave by the principle of dimin- ishing marginal utility. These curves summarize tradeoffs that an administrator would be willing to accept between the criteria. The problem is to deter- mine a point on the efficient frontier that also sits on the lowest of the indifference curves. This is the point of tangency between the two curves. Any alternative resulting in a number of cancelled surgeries and an average waiting time that yields a point that lies above the efficient frontier is a domi- nated alternative. Thus, Strategy 5, which opens the reserved beds to OT-emergency patients throughout the week, is dominated. The accompanying implica- tion, however, is that there is some efficient strategy that we have not explored here that would improve on that dominated alternative. For example, we might consider opening up the reserved beds to OT-emer- gency patients only on Friday, or opening up the reserved beds to A E and OT-emergency patients on Friday and Saturday. When no pure strategy can be found that improves on the ‘‘open-to-OT-emer- gency patients’’ alternative, the latter is still easily shown to be a dominated alternative. To show this we need only draw the line connecting the dots for pure Strategies 1 and 4. Comparing the numbers in Tables 6 and 8 reveals that all the DICU alternatives are dominated by some FBA scheme. The efficient frontier provides the administrator with a basis for evaluating any and all of the alternative FBA schemes. The latter might include some that involve additional costs, such as adding one bed to the unit, while still holding three in reserve for ES patients. Is that additional cost worth the gain from reaching a point below the efficient frontier? This question must be answered at the higher levels in the managerial hierarchy. The patient-care data to help management make that de- cision are provided in Table 10. The associated efficient frontier is provided in Fig. 6. With 14 beds, Strategy 4 is dominated. This is seen from the straight line connecting the dots for Strategies 3 and 5, which falls below the dot for Strategy 4. Comparing the data in Table 10 with their counterparts in Table 8 shows that the addi- tional bed only slightly reduces the overall bed-utili- zation rate and that all patients benefit from that bed. Under the current system, for example, the ES pa- tients benefit in that cancelled surgeries are almost halved. Other patients benefit in that the average waiting time is more than halved. Under Strategy 2 the additional bed in the non-reserved-bed section reduces cancelled surgeries by 44 and cuts almost an hour off the average queuing time. At a mini- mum, these data provide management food for thought. 4.2. SensitiÕity analysis A major advantage of FBA over a DICU in ameliorating the cancelled-surgery problem is its flexibility in that the number of reserved beds and the allocation scheme are readily altered without additional cost. An alteration can be implemented in a minimum of time and in response to such environ- mental changes as the patient mix and demand. One possible source of change that immediately leaps to mind is seasonal differences in the arrival rates from the different sources. As seen in Fig. 7, however, the Fig. 8. Impacts of increased arrival rates on the waiting time and the number of cancelled surgeries. 2-yr data from the hospital’s log book do not reveal any pronounced seasonal factor that affects overall demands. The comparable data for each of the four sources of referrals examined individually were equally unrevealing. We conducted sensitivity analyses to evaluate the impacts of the changes in patient arrival rates on the performance of FBA, increasing the patient arrival rates by 10 increments up to a 40 increase. By way of illustration, Table 11 presents the results for a 40 increase. While all numbers increased com- pared to the levels of Table 8, one standout result is that the impact on waiting time is largest for Strategy 2 and smallest for Strategy 5. The impact on can- celled surgeries is unchanged in that the proportions for the different methods are not affected by the change in arrival rates. This is shown by the graphs of Fig. 8. The bed-sharing FBA system, Strategy 5, is the most flexible of those considered here. This implies that greater flexibility improves performance when increased demand tightens the capacity con- straint.

5. Conclusions